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Morse and Lyapunov Spectra and Dynamics on Flag Bundles

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Morse and Lyapunov Spectra and Dynamics on Flag Bundles Morse and Lyapunov Spectra and Dynamics on Flag Bundles∗ Luiz A. B. San Martin† Lucas Seco‡ Instituto de Matem´atica Universidade Estadual de Campinas Cx.Postal 6065, 13.081-970 Campinas-SP, Brasil October 25, 2018 Abstract This paper studies characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G = KAN defines an additive cocycle over the flow with values in a = log A. Its Lyapunov exponents (lim- its along trajectories) and Morse exponents (limits along chains) are studied. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also that these sets are located in certain Weyl chambers, defined from the dynamics arXiv:0709.3278v1 [math.DS] 20 Sep 2007 on the associated flag bundles. As a special case linear flows on vector bundles are considered. AMS 2000 subject classification: Primary: 37B05, 37B35. Secondary: 20M20, 22E46. Key words: Morse exponents, Lyapunov exponents, Chain transitivity, semi- simple Lie groups, flag manifolds, Morse decomposition. ∗Research supported by FAPESP grant n0 02/10246-2 †Supported by CNPq grant n◦ 305513/03-6 ‡Supported by FAPESP grant n◦ 04/00392-7 1 1 Introduction In this paper we study characteristic exponents (Morse and Lyapunov vector spectra) associated to continuous flows evolving on principal bundles. The purpose is to relate these exponents to the dynamics on the flag bundles. We work with invariant flows on a general principal bundle whose structure group G is semi-simple or reductive. The objects to be considered are de- fined intrinsically from G and the principal bundle. Thus our characteristic exponents take values on a vector subspace a of the Lie algebra g of G, while the flag bundles in question have fiber F, a compact homeneous space of G (generalized flag manifolds of G). When G is a linear group a becomes a subspace of diagonal matrices and F a manifold of flags of subspaces. The vector valued spectra are defined via the cocycle over the flag bun- dle obtained by projection against an Iwasawa decomposition of G. They measure the exponential growth ratio of the flow the same way as the usual one-dimensional spectra measures the growth ratio of linear flows on vector bundles. (Actually the one-dimensional spectrum can be recovered from the vector valued one, via a representation of G, see Section 9.) This is the set up of Kaimanovich [16] to the proof of the multiplicative ergodic theorem of Oseledets. The Morse spectrum set is a concept of growth ratio along chains intro- duced by Colonius-Kliemann [5], [6] (see also Colonius-Fabri-Johnson [7] for a vector valued version). Each one of these sets depend on a chain component of a flow. In our case the Morse spectra are defined over chain components of the flow on an associated flag bundle. Our purpose here is to relate the geometry of the vector Lyapunov and Morse spectra with the geometry of the chain components on the flag bundle, which were studied in [4] and [22]. We better explain our results with the concrete example where G = Sl(3, R). Let X be a compact Hausdorff space endowed with a continu- ous flow (t, x) 7→ t · x (t ∈ T = Z or R) and suppose that φt (x, g) = (t · x, ρ (t, x) g) is a flow on Q = X × G with ρ a continuous cocycle taking values in G. Let A ⊂ G be the subgroup of diagonal matrices (positive en- tries) and N the subgroup of upper triangular unipotent matrices. Then the Iwasawa decomposition of G reads G = KAN with K = SO(3). For x ∈ X and u ∈ K we write ρ (t, x) u = kt (x, u) at (x, u) nt (x, u) ∈ KAN and look at the limit behavior as t →∞ of the matrix a (t, x, u) /t = log at (x, u) /t in the subspace a = log A. 2 This is done in combination with the asymptotics of the compact part kt (x, u). Via the action of G on K ≃ G/AN the component kt (x, u) becomes a flow on K and a (t, x, k) an additive cocycle over this flow. Thus the characteristic exponents defined by a depend on (x, k) ∈ X ×K. Nevertheless it is more convenient to avoid ambiguities and factor out the subgroup M ⊂ K of diagonal matrices with ±1 entries and get a (t, x, b) as a cocycle over the flow φF induced on X × F, where F ≃ K/M ≃ G/MAN is the manifold of complete flags in R3. Now let M be a chain component of the flow φF on X × F. Its associ- ated Morse spectrum set ΛMo (M) ⊂ a is defined by evaluating the cocycle a (t, x, b) on chains in M taking into account the jumps of the chains (see Definition 3.1 below). The Lyapunov spectrum set ΛLy (M) is the set of limits lim a (t, x, b) /t along the trajectories in M. By general facts ΛMo (M) is a compact convex set which contains ΛLy (M), which in turn contains the set of extremal points of ΛMo (M) (see Section 3 below). The chain components in X × F were described in [4], [22] with the as- sumption that the flow on X is chain transitive. The picture is the following: −1 There exists Hφ ∈ a and a continuous map x ∈ X 7→ fφ (x) = gxHφgx , gx ∈ G, into the set of conjugates of Hφ such that each chain component M has the form x∈X {x} × fix (fφ (x)). Here fix(fφ (x)) is a connected com- ponent of the fixed point set of the action on F of exp fφ (x) (the connected components canS be labelled accordingly so that they match each other). For example if the diagonal matrix Hφ has distinct eigenvalues λ1 > λ2 > λ3 with eigenvectors e1, e2, e3 then there are 3! = 6 isolated fixed points, namely the flag b0 = (span{e1} ⊂ span{e1, e2}) together with those obtained from b0 by permuting the basic vectors. A similar description holds for fφ (x), x ∈ X, allowing to label the fixed points, as well as the chain components, with the permutation group on three letters S3, which is the Weyl group W for Sl(3, R). In this case there are 6 distinct chain components denoted by Mw with w running over W = S3. The chain components are also labelled by the permutation group even if Hφ has repeated eigenvalues, but now some of the Mw can merge. In our Sl(3, R) example there are four possibilities for Hφ, namely (a) Hφ = diag(λ1 >λ2 >λ3) with six chain components Mw, w ∈ W. (b) Hφ = diag(λ1 = λ2 > λ3) with three chain components M1 = M(12), M(23) = M(123) and M(132) = M(13). 3 (c) Hφ = diag(λ1 > λ2 = λ3) with three chain components M1 = M(23), M(12) = M(132) and M(123) = M(13). (d) Hφ = diag(λ1 = λ2 = λ3) = 0 and the flow is chain transitive on X × F. (In all the cases M1 is the only attractor component and M(13) the only repeller. The equalities in the second and third cases are due to the cosets Wφw where Wφ is the subgroup of W fixing Hφ.) We say that the matrix Hφ is the block form of φ. It is determined by the dynamics on the flag bundles and is a combination of the parabolic type of the flow and of the reversed flow (see [4], [22] and Section 5 of the present article). The results of this paper provide a similar picture for the Morse and Lyapunov spectra over the chain components in terms of the Weyl group, the Weyl chambers and the block form Hφ of φ. For G = Sl(3, R) the roots of a are the functionals αij (diag(a1, a2, a3)) = ai − aj, for i =6 j ∈{1, 2, 3}. Their kernels are three straight lines containing the nonregular matrices in a, i.e., those having repeated eigenvalues. The complement to the lines is the union of 3! = 6 Weyl chambers where the Weyl group acts simply and transitively (see Figure 1). A chamber is determined by inequalities ai > aj > ak and we denote it by Cw if w ∈ W is the permutation sending (123) to (ijk). The main results of this paper prove the following estimates and symme- try properties of the vector valued Morse and Lyapunov spectra (see Figures 1 and 2): 1. The union w∈W ΛMo (Mw) of the several Morse spectra is invariant under the Weyl group. The same holds for the Lyapunov spectra S w∈W ΛLy (Mw). 2. TheS spectra ΛMo (M1) and ΛLy (M1) of the attractor component are invariant under the subgroup Wφ fixing Hφ. 3. Λ (M ) is contained in the interior of clC . This means that Mo 1 w∈Wφ w the multiplicities of the eigenvalues of any Morse exponent in Λ (M ) S Mo 1 do not exceed those of Hφ. The same estimate holds for ΛLy (M1), since ΛMo (Mw) is the convex closure of ΛLy (Mw). 4. ΛMo (M1) intercepts the closure of every chamber Cw, w ∈ Wφ. Hence there exists a Morse exponent in ΛMo (M1) whose eigenvalues have 4 Figure 1: Morse spectra for flows with G = Sl(3, R): block forms (a) and (b), each spectrum is labeled by its corresponding component.
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